Selecting an optimal approach to reduce energy crises under interval-valued intuitionistic fuzzy environment

The concept of interval-valued intuitionistic fuzzy sets is intellectually stimulating and holds significant utility in the representation and analysis of real-world problems. The development of similarity measures within the class of interval-valued intuitionistic fuzzy sets possesses significant importance across various academic disciplines, particularly in the fields of decision-making and pattern recognition. The utilization of similarity measures is of utmost importance in the decision-making process when implementing interval-valued intuitionistic fuzzy sets. This is due to its inherent capability to quantitatively assess the level of resemblance or similarity between two interval-valued intuitionistic fuzzy sets. In this article, the drawbacks of the existing similarity measures in the context of an interval-valued intuitionistic fuzzy environment are addressed, and a novel similarity measure is presented. Many fundamental properties of this new interval-valued intuitionistic fuzzy similarity measure are also established, and the effectiveness of this similarity measure is illustrated by presenting a useful example. Moreover, a comparison is given to demonstrate the validity of the newly proposed similarity measure within the existing knowledge of similarity measures in the interval-valued intuitionistic fuzzy environment. In addition, an algorithm is designed to solve multi-criteria decision making problems by means of the proposed measure in the interval-valued intuitionistic fuzzy setting. Furthermore, this newly defined similarity measure is successfully applied to select an optimal renewable energy source to reduce energy crises. Finally, we conduct a comparative study to showcase the authenticity of the recently defined technique within the existing knowledge of similarity measures in the interval-valued intuitionistic fuzzy environment.

Definition 2.5 17 Let I = [0, 1] and F : I 4 → R, be defined as: Then F meets the principles listed below.

Limitations of Current similarity measures of IVIFS
We demonstrate in the subsequent discourse that the similarity measures of IVIFS as delineate d i n 27,28,31,32 are i n e f fe c tu a l.L e t K = u, γ − K (u), γ + K (u) , σ − K (u), σ + K (u) |u ∈ U a n d � = u, γ − � (u), γ + � (u) , σ − � (u), σ + � (u) |u ∈ U be two IVIFS on U.  www.nature.com/scientificreports/Definition 3.1 27 Xu's and Chen similarity measure is defined as: The inefficiency of S 1 and S 2 is demonstrated by the following examples 3.2 and 3.3 respectively..60]} .The application of Definition 3.1 on the above IVIFS yields the following out- comes.S 1 (K, �) = S 1 (K, M) = 0.9 , where K = and K = M .This shows the indistinguishability characteristic of the similarity measure S 1 and hence is ineffective in this case.S 2 (K, �) = S 2 (K, M) = 0.9 , where K = and K = M .This shows the indistinguishability characteristic of the similarity measure S 2 and hence is ineffective in this case.Definition 3.4 28 Wei's et al. similarity measure is defined as: where, The invalidity of S W is illustrated by the following example 3.5 S W (K, �) = S W (K, M) = 0.21 , where K = and K = M .This shows the indistinguishability characteristic of the similarity measure S W and hence is ineffective in this case.Definition 3.6 Dhivya and Sridevi's similarity measure 31 is defined as: where The inefficacy of S D is illustrated by the following example 3.7.
Vol.:(0123456789) .60]} .The application of Definition 3.6 on the above IVIFS yields the following out- comes.S D (K, �) = S D (K, M) = 1.00 , where K = and K = M .This shows the indistinguishability characteristic of the similarity measure S D and hence is ineffective in this case.Definition 3.8 Luo and Liang's similarity measure 32 is given as: The incapability of S p is illustrated by the following example 3.9.

Proposed similarity measure
In this section, we introduce a novel similarity measure between IVIFSs and prove its structural properties.We also conduct a comparative study to demonstrate the validity of this newly defined similarity measure in comparison to the existing measures discussed in Section III.

Fundamental characteristics of proposed similarity measure on IVIFS
This section deals with the initiation of the novel similarity measure T between IVIFS and highlights its fundamental properties.
represent the membership and non-membership degree of IVIFS K respectively.In a simi- lar manner, γ − � (u), γ + � (u) and σ − � (u), σ + � (u) are the membership and non-membership degree of IVIFS respectively.The similarity measure T on K and is defined as follows: Herein, > 0. The validity of the similarity measure T is illustrated in the following example 4.2.1.T (K, �) = 0.7, and T (K, M) = 0.8 , where A = B and K = M .This shows the distinguishability charac- teristic of the similarity measure T and hence is effective in this case.
Theorem 4.1.3Let K, and M be any three IVIFSs defined on a universal set U = {u 1 , u 2 , u 3 , . . ., u n } and G be the class of all IVIFSs defined on U .Then, www.nature.com/scientificreports/admits all the structural properties of a similarity measure.
|u ∈ U be the interval valued intuitionistic fuzzy sets on U.
(S1) Case I First, we solve this property for T − which represents the lower membership and non-membership degrees of IVIFS K and . Since . The subsequent inequality results from considering Definition 2.5 (1): This implies that which further leads to Hence0 ≤ T − (K, �) ≤ 1 (1).Case 2: One can establish the above inequality for T + which represents the upper membership and nonmembership degrees of IVIFS K and .
As a result of this, we obtain: By comparing (1) and ( 2), we obtain the following inequality 0 ≤ T (K, �) ≤ 1. (S2) Case 1: First, we solve this property for the lower case T − Consider the following and Then we have In view inequality (1), T − (K, �) can be express as follows (2) This shows that Consequently, we obtain that if Case 2: One can establish the above equality for upper case T + Consequently, Furthermore, the condition (S3) is readily demonstrable.(S4) Case 1: First, we solve this property for the lower case By applying Definition 2.5 (2), the subsequent inequalities result: Then application of 4.1.1 in (3) gives that Further, we obtain the following: This implies that Then application of 4.1.1 in (3) and by adopting the above procedure we get By comparing the inequality ( 5) and ( 6), we get Case 2: One can establish above inequality for upper case T + .

Comparative analysis
In the subsequent discourse, we establish a comparison with pre-existing similarity measures in the IVIFS setting in order to illustrate the reliability and practicability of the suggested similarity measure.In Table 1, upon comparing the both columns of the known similarity measures S 1 , S 2 , S w and S P , it becomes evident that these similarity measures are not reasonable as they do not satisfy the following relation.S(K, �) = S(K, M) , while and = M indicating that the existing similarity measures are not reasonable.Moreover, S D (K, �) = S D (K, M) = 1.00 while = M , showing that t S D fails to meet the condition (S3) of Definition 2.4.However, the proposed similarity measure T effectively addresses all these cases.

Utilization of the proposed IVIF similarity measure in MCDM context
This section provides a method to address MCDM issues in order to demonstrate the significance of the suggested similarity measure T within the IVIF context.
Denote the collection of different alternatives as {K 1 , K 2 , K 3 , . . ., K m } and denote the collection of attributes as X = {ǫ 1 , ǫ 2 , . . ., is an m × n IVIF decision matrix, where represent a test sample as categorized by the IVIFS.The method that is utilized to resolve the MCDM challenge within the IVIF paradigm is constructed as follows: Step 1.
Convert the decision matrix M into a normalized matrix S = s ij m×n (if necessary), where S ij is computed by the following equation: Step 2. Calculate the similarity measures T (K i , �) between K i , where i = 1, 2, 3, . . ., m, and as follows: Step 3. Rank each alternative and select the one with the highest value.
; for loss type criteria Energy is necessary for our continued survival and plays a role in almost every aspect of our lives.If we did not have access to energy, we would be considerably limited in our ability to enjoy the conveniences and luxuries of modern life.This is due to the fact that the definition of energy insecurity changes depending on whether a country meets its energy requirements through self-production of energy, imports of energy from other countries, or exports energy to other countries to fulfill its needs.Energy security can be conceptualized as the state of perpetually possessing readily available, economically viable, and accessible energy.This phenomenon is observed in nations that have achieved full economic autonomy.The limited availability of readily available electricity has caused a range of complications that extend throughout Asia, including Pakistan.Pakistan's energy sector is currently facing a crisis due to a significant shortfall in meeting the increasing energy demands that have accumulated over the past few decades.The demand for energy is increasing rapidly as a result of population growth, urbanization, and industrialization, but the supply of traditional energy sources is insufficient.The energy shortage has resulted in frequent power blackouts, which have hindered economic development, disrupted everyday routines, and obstructed technical advancements.The energy crisis, which is the main cause of economic depletion, has a significant impact on Pakistan's economy.The current crisis originated from a shift in fuel composition that occurred twenty years ago, during which electricity generation increasingly depended on imported furnace oil rather than hydropower.The rise in power generation costs, along with the significant line losses, has necessitated tariff expeditions, resulting in financial losses for power generation, transmission, and distribution corporations.On the other hand, the widespread concern that the world's fossil fuel sources will run out in the near future and that the price of energy will continue to slowly rise is a key issue in today's world.Developing countries may finally be able to find a solution to their long-standing energy problems with the help of renewable energy sources and technology.Given the enormity of Pakistan's present energy challenge, switching to renewable energy sources may prove to be the most time-and cost-efficient way to solve the problem.
Renewable energy sources are inexhaustible and beneficial to the environment.Biofuels (e.g., ethanol, biodiesel), geothermal energy (by harnessing the thermal energy of water or vapor to drive turbines that generate electricity), organic matter (e.g., dung, wood, vegetation), wind, and oceanic waves are all examples of renewable energy sources.The implementation of renewable energy sources is critical in the prevention and management of energy crises.The significance of renewable energy lies in its capacity to meet the growing demand for electricity while preventing the depletion of finite natural resources.Reduced reliance on foreign fuels also reduces the potential for environmental problems like gasoline spills and emissions.Our long-term energy needss could be met by renewable sources, provided we have enough of them and use a variety of fuels.In this article, we propose a step-by-step procedure for choosing the best renewable energy source by using the newly defined IVIFS similarity measure.Let {E 1 , E 2 , E 3 , E 4 , E 5 , E 6 } be the renewable energy sources for electricity generation.A decision making problem using the newly proposed IVIFSs similarity measure is analyzed to assess the six renewable energy sources.Let be an IVIFS classified as test sample to evaluate the performance of a specific renewable energy source.The flowchart of the MCDM problem is illustrated in the Figure 1.
In order to address the MCDM problem, we opt for a unique alternative.Next, we identify and consider all the variables that could potentially influence these different choices, and we create a decision matrix.Subsequently, we select a test sample.Once all the input elements have been selected, we proceed to apply the proposed similarity measure to each alternative using the test sample.Ultimately, we choose the alternative with the highest value, which is considered our optimal choice.
The information given by decision maker for the above six renewable energy sources are evaluated under IVIF environment and are summarized in Table 2.The normalized matrix is presented in the subsequent Table 3.
Step 2. The similarity measure of each alternative E i corresponding to the set computed by using Definition 4.1.1.and is given by.

Limitations of the current study
In the context of MCDM, the method proposed in this work is subject to a number of limitations, despite its enticing advantages.These limitations become apparent when the total score of membership and non-membership is greater than 1 or when neutral membership is involved.These shortcomings can be addressed by using IV Pythagorean and IV picture fuzzy scenarios.The IV Pythagorean and IV picture fuzzy sets have the potential to address the limitations of IVIFS by offering better uncertainty representation, greater aggregation, accessibility, particular modifications, and more rigorous mathematical theory.Furthermore, our proposed approach primarily addresses one-dimensional issues.To handle scenarios involving 2-D information about a physical phenomenon, the utilization of the complex IVIF approach becomes an effective option.

1 .
E 1 : Tidal energy 2. E 2 : Wind energy 3. E 3 : Solar energy 4. E 4 : Hydropower energy Choose the distinct alternatives Choose the attribute relative to alternatives Choose the test sample to select the best renewable energy Input Compute the similarity measure of each renewable energy source Select the best alternative having the maximum value Output

Figure 1 .
Figure 1.Step-by-step procedure for choosing the best renewable energy source.

Table 1 .
Comparison of similarity measures within the framework of IVIFS.

the best renewable energy source to reduce energy crises
. E 5 : Geothermal energy 6.E 6 : Biomass energy Let X = {ǫ 1 , ǫ 2 , ǫ 3 , ǫ 4 } be the criterion use to evaluate the efficacy of various renewable energy sources.